(*
  DaoMath Number System Construction
  从 ±0 构造数系
*)

Require Import Reals.
Require Import Arith.
Require Import InnateBalance.
Require Import DualPair.
Open Scope R_scope.

(** * 自然数构造 *)

(** 自然数来自后继迭代 *)
Fixpoint nat_from_balance (n : nat) : R :=
  match n with
  | O => 0
  | S n' => 1 + nat_from_balance n'
  end.

(** 定理: 自然数构造保持 Peano 公理 *)
Theorem nat_zero : nat_from_balance 0 = 0.
Proof. reflexivity. Qed.

Theorem nat_successor : forall n,
  nat_from_balance (S n) = 1 + nat_from_balance n.
Proof. intros n. reflexivity. Qed.

(** 定理: 自然数加法同态 *)
Theorem nat_addition_homomorphism : forall n m,
  nat_from_balance (n + m) = 
  nat_from_balance n + nat_from_balance m.
Proof.
  intros n m.
  induction n as [|n' IH].
  - simpl. lra.
  - simpl. rewrite IH. lra.
Qed.

(** * 整数构造 *)

(** 整数作为自然数的对偶对 *)
Definition Z_from_balance (z : Z) : R :=
  match z with
  | Z0 => 0
  | Zpos p => INR (Pos.to_nat p)
  | Zneg p => - INR (Pos.to_nat p)
  end.

(** 定理: 整数对偶性 *)
Theorem integer_duality : forall p : positive,
  Z_from_balance (Zneg p) = dual (Z_from_balance (Zpos p)).
Proof.
  intros p.
  unfold Z_from_balance, dual. simpl.
  reflexivity.
Qed.

(** 定理: 整数加法保持对偶 *)
Theorem integer_addition_dual : forall z1 z2 : Z,
  Z_from_balance (Z.opp z1 + Z.opp z2) = 
  dual (Z_from_balance (z1 + z2)).
Proof.
  intros z1 z2.
  (* 证明需要更多引理 *)
Admitted.

(** * 有理数构造 *)

(** 有理数作为整数对 *)
Record Q_from_balance : Type := mkRational {
  numerator : Z;
  denominator : positive;
}.

(** 有理数到实数的嵌入 *)
Definition Q_to_R (q : Q_from_balance) : R :=
  Z_from_balance (numerator q) / INR (Pos.to_nat (denominator q)).

(** 定理: 有理数乘法逆元存在 *)
Theorem rational_multiplicative_inverse :
  forall q : Q_from_balance,
  numerator q <> Z0 ->
  exists q_inv : Q_from_balance,
  Q_to_R q * Q_to_R q_inv = 1.
Proof.
  intros q H_nz.
  exists (mkRational (Zpos (denominator q)) 
                     match numerator q with
                     | Zpos p => p
                     | Zneg p => p
                     | Z0 => xH (* 不会到达 *)
                     end).
  unfold Q_to_R. simpl.
  (* 详细证明需要分情况 *)
Admitted.

(** * 完备化：从有理数到实数 *)

(** Cauchy 序列（抽象） *)
Parameter CauchySeq : Type.
Parameter cauchy_limit : CauchySeq -> R.

(** Cauchy 序列收敛公理 *)
Axiom cauchy_convergence : forall (seq : CauchySeq) (eps : R),
  eps > 0 ->
  exists N : nat, forall n m : nat,
  (n >= N)%nat -> (m >= N)%nat ->
  Rabs (cauchy_limit seq - cauchy_limit seq) < eps.

(** * 复数构造 *)

Record Complex : Type := mkComplex {
  real_part : R;
  imag_part : R;
}.

(** 虚数单位 *)
Definition i : Complex := mkComplex 0 1.

(** 复数加法 *)
Definition C_plus (z w : Complex) : Complex :=
  mkComplex 
    (real_part z + real_part w)
    (imag_part z + imag_part w).

(** 复数乘法 *)
Definition C_mult (z w : Complex) : Complex :=
  mkComplex
    (real_part z * real_part w - imag_part z * imag_part w)
    (real_part z * imag_part w + imag_part z * real_part w).

(** 定理: i² = -1 *)
Theorem i_squared : C_mult i i = mkComplex (-1) 0.
Proof.
  unfold i, C_mult. simpl.
  f_equal; lra.
Qed.

(** 复数的对偶：共轭 *)
Definition C_conjugate (z : Complex) : Complex :=
  mkComplex (real_part z) (- imag_part z).

(** 定理: 共轭的共轭是自身 *)
Theorem conjugate_involutive : forall z : Complex,
  C_conjugate (C_conjugate z) = z.
Proof.
  intros z.
  unfold C_conjugate. simpl.
  destruct z. simpl. f_equal. lra.
Qed.

